Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

234 CHAPTER 8 | Introduction to Hypothesis Testing
technique. In each case, keep in mind that the z-score serves as a general model for other
test statistics that will come in future chapters.
The z-Score Formula as a Recipe The z-score formula, like any formula, can be
viewed as a recipe. If you follow instructions and use all the right ingredients, the formula
produces a z-score. In the hypothesis-testing situation, however, you do not have all the
necessary ingredients. Specifically, you do not know the value for the population mean
(μ), which is one component or ingredient in the formula.
This situation is similar to trying to follow a cake recipe where one of the ingredients is
not clearly listed. For example, the recipe may call for flour but there is a grease stain that
makes it impossible to read how much flour. Faced with this situation, you might try the
following steps.
1. Make a hypothesis about the amount of flour. For example, hypothesize that the
correct amount is 2 cups.
2. To test your hypothesis, add the rest of the ingredients along with the hypothesized
flour and bake the cake.
3. If the cake turns out to be good, you can reasonably conclude that your
hypothesis was correct. But if the cake is terrible, you conclude that your
hypothesis was wrong.
In a hypothesis test with z-scores, we do essentially the same thing. We have a formula
(recipe) for z-scores but one ingredient is missing. Specifically, we do not know the value
for the population mean, μ. Therefore, we try the following steps.
1. Make a hypothesis about the value of μ. This is the null hypothesis.
2. Plug the hypothesized value in the formula along with the other values (ingredients).
3. If the formula produces a z-score near zero (which is where z-scores are supposed
to be), we conclude that the hypothesis was correct. On the other hand, if the
formula produces an extreme value (a very unlikely result), we conclude that the
hypothesis was wrong.
The z-Score Formula as a Ratio In the context of a hypothesis test, the z-score formula
has the following structure:
z 5 M 2m
s M
sample mean 2 hypothesized population mean
5
standard error between M and m
Notice that the numerator of the formula involves a direct comparison between the sample
data and the null hypothesis. In particular, the numerator measures the obtained difference
between the sample mean and the hypothesized population mean. The standard error in the
denominator of the formula measures the standard amount of distance that exists naturally
between a sample mean and the population mean without any treatment effect that causes the
sample to be different. Thus, the z-score formula (and many other test statistics) forms a ratio
actual difference between the sample sMd and the hypothesis smd
z 5
standard difference between M and m with no treatment effect
Thus, for example, a z-score of z = 3.00 means that the obtained difference between the
sample and the hypothesis is 3 times bigger than would be expected if the treatment had
no effect.